Sound and vibration phenomena often are analysed in order to optimise them, e.g. to reduce the amount of disturbance that is generated for a user or the environment, to make a product compliant with specifications or with regulations set or to identify sources of disturbance, to control a system, etc. A known way for sound and vibration analysis is making use of order tracking. Order tracking typically is applied to variable-speed systems generating sound or another mechanical vibration.
Order tracking concerns the extraction of the complex envelope of order components from measured mechanical and/or acoustic vibrations. Orders are harmonic components of which the kernel frequency is a multiple or fraction of the angular speed of the periodic excitation.
The oldest order tracking technique is based on performing Fourier Transforms on time domain data. Fourier Transforms with constant kernel frequencies are used. The transformed data is displayed in either a waterfall or color map format. Orders of interest are then estimated through determining the average frequency of each order over which the Fourier Transforms were performed and extracting the corresponding frequency lines. The limitations of these techniques are many and can be significant. The two largest limitations are limited order resolution at lower rotational speeds and slow sweep rates.
Considerable improvements in order tracking were achieved since the late eighties. DC-estimation techniques were developed in which the Fourier Transform kernels explicitly take account of the changes in rotational speed. Two variants of DC-estimation methods exist, i.e. DC estimation in the angle-domain and DC estimation in the time-domain. The angle-domain variant, also referred to as resampling-based order tracking, is for example known from U.S. Pat. No. 6,351,714. The method is based on a limited observation interval of the angle-sampled signal x(α), during which the complex order component envelopes Xk(ρ(α)), being a function of the rotational speed ρ which is a function of the angle α, are assumed to be constant. An interval [θ−Δα, θ+Δα] is picked wherein x(α) is periodic with period Q and wherein the variation of the angle Δα=Qπ. 1/Q is also known as the order resolution. The order is then estimated as follows:
                                                        X              ^                        k                    ⁡                      (                          ρ              ⁡                              (                θ                )                                      )                          =                              ∫                          θ              -              Δα                                      θ              +                              Δ                ⁢                                                                  ⁢                α                                              ⁢                                    C              w                        ⁢                          W              ⁡                              (                α                )                                      ⁢                          x              ⁡                              (                α                )                                      ⁢                          ⅇ                                                j                  ⁢                                                                          ⁢                  k                  ⁢                                                                          ⁢                  α                                Q                                      ⁢                          ⅆ              α                                                          [        1        ]            wherein W(α) is a windowing function to avoid leakage and Cw is a window correction factor. By moving the independent variable to the time-domain, we obtain the time-domain variant, which is mathematically equivalent and often referred to as time-variant DFT. The time-variant DFT formulation is as follows:
                                                        X              ^                        k                    ⁡                      (                          ρ              ⁡                              (                T                )                                      )                          =                              ∫                          T              0                                      T              1                                ⁢                                    C              w                        ⁢                          W              ⁡                              (                                  α                  ⁡                                      (                    t                    )                                                  )                                      ⁢                          x              ⁡                              (                t                )                                      ⁢                          ⅇ                                                j                  ⁢                                                                          ⁢                  k                  ⁢                                                                          ⁢                                      α                    ⁡                                          (                      t                      )                                                                      Q                                      ⁢                                          ⅆ                α                                            ⅆ                t                                      ⁢                          ⅆ              t                                                          [        2        ]            where:α(T)=0α(T0)=θ−Δαα(T1)=θ+Δα
The weakness of the DC-estimation approach is the assumption that the order Xk(ρ(α)) must be constant over the observation interval. It assumes a zero angle-domain order bandwidth, i.e. Bαk (1/rad)=0. However, this assumption is only true at constant rotational speed. When the rotational speed ρ changes in the observation interval, which is obviously the case in run-up and coast-down measurements, Bαk is not longer zero and the DC-estimation method starts suffering from order crosstalk. Orders then leak into adjacent ones and cannot be longer separated. The cross-talk and resulting DC-estimation errors increase with (i) decreasing order spacing 1/Q, (ii) decreasing rotational speed ρ, (iii) increasing angular acceleration dρ/dt and (iv) increasing order bandwidth Bρk (s/rad) in the rotational speed domain. Here, Bρk characterizes the order envelope variations with rotational speed. Bρk is a system characteristic which depends on the system transfer function characteristics.
Another known and widely-used type of order tracking technique is referred to as a Vold-Kalman order tracking approach for rotating machinery. This time-domain method centres the order of interest about DC and applies a particular type of low-pass filter to the phasor-shifted data. The Vold-Kalman order tracking filter acts as an autoregressive, IIR type of filter with a limited number of poles. The tracking characteristics of the filter are determined by the HCF (Harmonic Confidence Factor) weighting parameter. Undesired phase distortions on the order estimates are minimized by adopting a total Least Squares solution algorithm. This algorithm estimates the full order envelope at once from the complete data signal. However, this makes the Vold-Kalman approach computationally very heavy and explains its off-line character and use.
There is a further need for good methods and systems for order tracking, being accurate and at the same time computational efficient.